Acceleration and weight

I don't understand conceptually how something accelerating down weighs less. For example, the elevator example of course if you have something with a string and a weight and the acceleration of the elevator increases upward as positive, you get

T - mg - ma = 0

and for acceleration down, you get

T - mg + ma = 0

This to me doesnt make sense, because shouldnt the equations be switched around because ma is + going up and - going down??

If you are measuring acceleration as postive upwards, then "g" has a negative value.

Once you see that, the equations are correct. For eaxmple if the object is in free fall, a = g = -9.8 m/s^2 and T = 0.

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woops I meant down. The example I think of is how you weigh less when driving down a large hill and your groin gets that funny feeling. How does this work if the gravity and acceleration are going in the same direction, shouldnt they two accelerations add, and then when you multiply by the mass the weight would be bigger??

In the groin/car example, the car is accelerating down away from you and your groin (sure I could switch back to the elevator example, but I like this one better). If the car were stationary, gravity would be accelerating you down, and the car would just sit there. You would feel the equal and opposite force from the seat. When the car accelerates away from you, you feel a slightly lower reaction force.

You are correct... if the weight is being measured on your back, but it probably wasn't being measured there. The car may be a little harder to imagine because it is 2D rather than 1D like the elevator.

How does this work if the gravity and acceleration are going in the same direction, shouldnt (the) two accelerations add, and then when you multiply by the mass, the weight would be bigger??

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Lets imagine this in an elevator... no numbers.
The elevator is stationary. Gravity pulls on both you and the elevator, but the elevator is held stationary by cables or some sort of magic. The elevator exerts a force upward on you that is equal and opposite the force that you exert on the elevator due to gravity pulling you down. So the weight that would be measured is the force between you and the elevator from gravity pulling you down and the elevator pushing you up.

If the elevator accelerates down, gravity stays the same (unless it is a really big elevator). The force up from the elevator will be less since it is now accelerating away from you. The acceleration of the elevator adds changes from 0 to some amount downward. The upward force on you exerted by the elevator will be less since it is now accelerating away from you. The force upward will be

The force that would be exerted to counter gravity, minus the acceleration of the elevator.

(yes I'm mixing acceleration and force, so this is not a precise mathematical expression)

Does that make more sense? If the elevator were to accelerate enough, you would hit the ceiling and if you move the scale from the floor to the ceiling, your weight would increase if the elevator accelerates more. Since weight is a measure of force, you could treat it as a vector quantity... but I think it is probably defined as the force downward.